Despite the fact that the giant shadow of his father, the legendary movie mogul Darryl F Zanuck, loomed large over him for most of his life, Richard Zanuck, who has died of a heart attack aged 77, triumphantly overcame inferences of nepotism and wisecracks such as "the son also rises", to become one of the most successful Hollywood producers in the last 50 years. His reputation was due initially to Jaws (1975), among the highest grossing movies up to that time, and he was a key figure in launching the career of its director, Steven Spielberg. Zanuck was Oscar-nominated for Jaws and won the Academy Award for best picture with Driving Miss Daisy (1989).

Born in Los Angeles, Zanuck seemed destined to enter show business. He was the third child and only son of the co-founder and head of 20th Century Fox, and the silent-film star Virginia Fox, best known for co-starring with Buster Keaton in several shorts.

While Zanuck was studying at Stanford University, his father got him a job in the story department at Fox. In 1956, he was appointed vice-president, in charge of the US operations, of Darryl F Zanuck Productions, his father's newly formed independent production company. The following year, he assisted his father on two costly, starry, shot-on-location vehicles, The Sun Also Rises and Island in the Sun.

Then, in 1959, the elder Zanuck allowed the younger off the leash to produce Compulsion, a modest, black-and-white picture, directed by Richard Fleischer. Based on the notorious 1924 murder case of the thrill-killers Leopold and Loeb, this stylish and tense courtroom drama, with a flamboyant extended cameo by Orson Welles, proved that Zanuck could bring a film in within budget and on time, as well as gaining it critical and public approval. Unfortunately, his second production, Sanctuary (1961), adapted from two potboiler novels by William Faulkner, was a turgid southern melodrama to which Zanuck unaccountably assigned Englishman Tony Richardson as director.

In 1962, when Darryl was having troubles with Fox on Cleopatra and The Longest Day, he offered The Chapman Report to Warner Bros with his son as sole producer and George Cukor as director. However, Warners insisted that Darryl co-produce the movie, which turned out to be a rather silly and sleazy examination of the psychosexual problems of four women in Los Angeles. Nevertheless, it made money.

Richard was appointed vice-president in charge of production and later president of Fox. Huge hits including The Sound of Music (1965) were made under his watch, but he also greenlighted expensive flops such as Doctor Dolittle (1967) and Star! (1968). In 1970, he was sacked by his father, who was still an executive and main shareholder at Fox. "It was hard to see my father go through that," he remarked. "I recovered from being fired, I don't think he ever did." In fact, it enabled Zanuck to finally become a producer in his own right.

In 1972, Zanuck linked up with David Brown to form Zanuck/Brown Productions, which released films through Universal Pictures. It was Zanuck and Brown who decided to give the 26-year-old Spielberg the chance to direct his first feature only on the evidence of his TV movies. The Sugarland Express (1974), a pleasing combination of character-driven true-life tale and screwball chase, was a modest success.

Even before its release, The Sugarland Express convinced Zanuck and Brown that Spielberg could be entrusted with Jaws. They had first come across Peter Benchley's novel after Brown's wife, Helen Gurley Brown, editor-in-chief of Cosmopolitan magazine, came across a note from the magazine's books editor that it "might make a good movie". Zanuck and Brown read the book in a single night and immediately agreed that it was the most exciting thing they had ever read and that they wanted to produce a movie version. The rights were then bought for $175,000 (£112,350), and the film was given an estimated budget of $3.5m. (It rose to $9m.) The editor Verna Fields, who was awarded one of the film's three Oscars, was advised by Zanuck to rework some of the material in post-production. According to Zanuck: "She actually came in and reconstructed some scenes that Steven had constructed for comedy and made them terrifying, and some scenes he shot to be terrifying and made them comedy scenes."

Jaws was a hard act to follow, but Zanuck and Brown had hits with MacArthur (1977), a biopic of the rebel second world war general, played by Gregory Peck; The Verdict (1982), Sidney Lumet's intelligent study of an alcoholic lawyer (Paul Newman); and the blockbuster Cocoon (1985), which was co-produced by Zanuck's third wife, Lili Fini, whom he married in 1978.

Together, Zanuck and his wife produced Driving Miss Daisy, which won four Academy Awards, including best picture, a prize which tradition demands is always handed to the producer, who may or may not have contributed to the quality of the film. In this case, Zanuck had been instrumental in the selection of Bruce Beresford as director, although the Oscars host Billy Crystal referred to Driving Miss Daisy as the movie that "directed itself" – alluding to Beresford's absence from the list of nominated directors. In a sense, the film did direct itself because all the right ingredients were there from the start. This tender picture, dealing with friendship and race relations, played by Jessica Tandy and Morgan Freeman with charm, humour and authority, transcended the rather old-fashioned concept of a mistress and a dignified black servant respectable enough to know his place.

Zanuck later co-produced six of Tim Burton's films, from Planet of the Apes (2001) to Dark Shadows (2012).

With his first wife, Lili Gentle, Zanuck had two daughters, Virginia and Janet. He had two sons, Harrison and Dean, with his second wife, Linda Harrison. Both of those marriages ended in divorce. He is survived by Lili Fini and his children.

• Richard Darryl Zanuck, film producer, born 13 December 1934; died 13 July 2012